Mr Blinken said he would speak directly to Foreign Minister Sergey Lavrov in the coming days to discuss the proposed deal, in what would be their first conversation since Russia invaded Ukraine five months ago.
“We put a substantial proposal on the table weeks ago to facilitate their release,” Mr Blinken said.
“Our governments have communicated repeatedly and directly on that proposal and I’ll use the conversation to follow up personally and I hope move us towards a resolution.”
He did not divulge the details of the proposal but emphasised that President Joe Biden “was not only directly involved”, and “he signs off on any proposal that we make, certainly when it comes to Americans who are being arbitrarily detained abroad”.
The Biden administration has declined to comment on whether it would agree to release Viktor Bout, a Russian arms dealer serving a 25-year prison sentence in the US, as part of a swap for Griner and Mr Whelan.
But CNN reported on Wednesday that Washington would release Bout in exchange for the release of Griner, a WNBA player, and Mr Whelan, a former marine.
The US had previously spurned the possibility of an exchange to free Bout.
Nicknamed the “Merchant of Death”, Bout was in 2012 sentenced 25 years in prison in Illinois on charges of conspiring to kill Americans, exporting anti-aircraft missiles and supporting a terrorist organisation. Russia has denied those charges.
“I can’t and won’t get into any of the details of what we proposed to the Russians … My hope would be, in speaking to Foreign Minister Lavrov, I can advance the efforts to bring them home,” Mr Blinken said.
Mr Whelan was detained in Russia in December 2018 on spying charges and was sentenced to 16 years in prison. The US government strongly denies the charges.
Griner was detained in February on drug-trafficking charges after cannabis oil was found in her bag at a Moscow airport.
She pleaded guilty to the charges this month although her defence team has said the vape oil was for medicinal purposes. She faces up to 10 years in prison.
Mr Blinken said he also planned to speak with Mr Lavrov about the UN-brokered deal to free Ukrainian grain from storage and possible Russian attempts to annex portions of eastern and southern Ukraine.
“If we can help through our direct diplomacy, for the Russians to make good on the commitments they've made, that will help people around the world, then I'm determined to do it,” Mr Blinken said.